Calibrated Multivariate Regression with Application to Neural Semantic Basis Discovery

TL;DR

This paper introduces CMR, a calibrated multivariate regression method that improves high-dimensional model fitting by adjusting for noise levels, with theoretical guarantees, efficient algorithms, and practical applications including brain activity prediction.

Contribution

The paper presents CMR, a novel calibrated regression method that enhances performance and tuning insensitivity in high-dimensional multivariate regression models.

Findings

CMR outperforms existing methods in simulations.

CMR achieves optimal convergence rates.

CMR is effective in brain activity prediction.

Abstract

We propose a calibrated multivariate regression method named CMR for fitting high dimensional multivariate regression models. Compared with existing methods, CMR calibrates regularization for each regression task with respect to its noise level so that it simultaneously attains improved finite-sample performance and tuning insensitiveness. Theoretically, we provide sufficient conditions under which CMR achieves the optimal rate of convergence in parameter estimation. Computationally, we propose an efficient smoothed proximal gradient algorithm with a worst-case numerical rate of convergence $\cO(1/\epsilon)$, where $\epsilon$ is a pre-specified accuracy of the objective function value. We conduct thorough numerical simulations to illustrate that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR to solve a brain activity prediction problem and find that it is as competitive as a handcrafted model created by human experts. The R package \texttt{camel} implementing the proposed method is available on the Comprehensive R Archive Network \url{http://cran.r-project.org/web/packages/camel/}.